In lessons 8 – 10 the focus was on modeling functional relationships with proportional functions, linear functions, and piecewise linear functions. In addition to providing students practice with the important skill of writing equations for these types of functions, students were repeatedly asked to make connections between graphs and the real life situations they model.
Throughout the remainder of the unit, students will begin working on extending the learning about volume from Grade 7, where they learned the formula for the volume of a right rectangular prism. Although this is not a priority standard for the grade, it is important in the progression of student learning and will be tested in state testing here in California, which is an SBAC state. In addition, students will continue to be given opportunity to explore linear and nonlinear function relationships in tables and graphs.
At this point in the year, many teachers are feeling behind and wondering about what they can condense to help them finish all the content they are responsible to teach before the end of the year. Non-priority standards seem like a great place to look for cuts. I will point out a few condensing points in this walk through.
Lesson 11 Filling Containers
11.1 Which one doesn’t belong.
Take this opportunity to refresh/introduce vocabulary students will use during this half of the unit by creating a list of vocabulary for geometric solids as students explain their thinking.
11.2 Height and volume
Because the textbook suggests using a lab setting with graduated cylinders that not everyone has access to, I know some teachers consider skipping this portion of the lesson. If you typically use print resources and slides from Open Up Resources community resources you may not notice there is also an awesome digital application available for students to explore the relationship between volume and height. Interacting with lab materials or this app will ensure that all of your students have a solid conceptual understanding of volume before we begin working on formulas.
Spend the time to do a thorough activity synthesis here, even if it means you skip 11.3. You can get the essential math learning here.
11.3 What is the Shape
If time allows, this is a nice next step in volume explorations which connects the modeling from lessons 9 and 10 to our volume explorations. The Desmos activity Waterline goes great with this activity and could be used any time after this lesson if you include activity 11.3 in your student’s learning experiences.
11.4 Which Cylinder?
A great cool down and follows nicely from the activity synthesis of 11.2.
Lesson 12 How Much Will Fit?
12.1 and 12.2 tap into the fun and challenge of Estimation 180 type activities to:
- help students think about volume,
- expose them to different shaped containers we will be finding volumes for this unit
- Practice using correct academic language to describe these solids
- Think about units of measure appropriate to volume
- Create a curiosity about how we might calculate volume of solids that are not rectangular prisms.
Keep the activities fun and light.
12.3 Do you know these figures?
This connects back to the Which one doesn’t belong? from 11.1. Students learn/practice more with academic vocabulary and learn to draw these figures on their papers.
The Lesson Summary is a good time to have students make notes about what they have learned so far about geometric solids. Have students do an individual write for these, then use a stand and talk to have them share and add on to these notes. Close with a class discussion.
Extra time? Work ahead to do 13.1 ( a review of 7th grade circle work) This will ease the time crunch for lesson 13, which is a key lesson in this portion of the unit.
Tight on time? 11.1, 11.2, 12.3, and 11.4 make a nice single combined lesson if you are feeling behind at this point.
(11.3 could be pulled back right before state testing to review several topics)
Lesson 13 The Volume of a Cylinder (A full day with nothing to skip)
13.1 A Circle’s Dimensions
This warmup is meant to bring up students 7th grade learning about circles including words like radius, diameter, the number pi, and how to calculate a circle’s area. Don’t skip the launch, which reviews many of these things. If no student can come up with the formula for the area of a circle, or if students can produce formulas for both area and circumference, this is a great time to model using digital resources, including asking Siri, Google, or Alexa.
13.2 Circular Volumes
This activity connects previous learning about volume of prisms to the new volumes they will be learning. Spend the time to be sure they get this. A physical models (using unifix cubes, stacking boxes or stacking cylinders) are extremely helpful to students visualizing this learning.
13.3 A Cylinder’s Dimensions
This doesn’t take long, and is more important than you realize. Circulate and catch errors as students try to sketch the radius and height for each of these. Watch on D and E for students labeling the diameter as the height. In activity synthesis, discuss this confusion.
13.4 A Cylinder’s Volume/ 13.5 Liquid Volume
If you end up getting sucked into doing 13.4 together as a class, both the cooldown and practice problem number 1 give immediate chances for students to practice and apply this learning. Be sure to make time for at least one of those before students leave your room.
Lesson 14: Finding Cylinder’s Dimensions
14.1 A Cylinder of Unknown Height
This is a great set for the rest of the lesson. I lie adding the questions “If I told you the height was 3, could you figure out the volume?” If I told you the volume was 32pi , could you figure out the height?”
14.2 What’s the Dimension?
Use this to discuss methods students use to find the missing dimensions. The curriculum suggests using the Math Language Routine “Stronger and Clearer” to focus the attention on refining their explanations. For partners or groups that finish quickly, the Are you ready for more is a nice extension. As you circulate, keep a paper with possible answers handy for those groups, so you can focus your time on those struggling with their explanations of #1 and 2. Since there are an infinite number of solutions to the Are you ready for more, all students should be able to continue working until you are ready to lead the synthesis.
14.3 Cylinders with Unknown Dimensions
For those who are always feeling the need for more student practice, here’s a great opportunity for students to develop fluency with the formula work. Don’t get trapped into feeling that all students must finish every line of this table during class. After all students have finished the first 4 rows, they have done enough thinking to be able to follow along with the discussion. You may need to project a completed table to allow all students to participate in a discussion of patterns that emerged in the table. (see activity synthesis questions in the teacher materials).
14.4 Find the Height
Take time to have students complete individually and turn in this cooldown. Use it to evaluate how students are doing on their work with cylinders and address common mistakes or misconceptions the next day.
Lesson 15 The Volume of a Cone
15.1 Which has a larger volume?
The estimations in #1 and 2 will be answers in activity 15.2. Now is the time to get students to invest in their guesses, as well as help them be able to successfully sketch a cone.
15.2 From Cylinders to Cones
If you have a geometric solids set like these, testing out students’ estimations from activity 1 can be great fun. Usually students think it will take 2 cones to fill one cylinder with the same base, and the fact that it takes three ( and only a few students got that correct) makes the formula especially memorable.
If you do not have a set of solids, a nice video is included in the materials. You can watch it here.
The questions in this activity mirror ones students will see on the unit assessment so take time to fully synthesize this, recording the formula for the volume of a cone on classroom anchor charts and in student notes.
15.3 Calculate that Cone/ Are you Ready for More? /15.4 Cool-down
Here is an opportunity for students to put this learning into practice. Be sure to include time for the compound solid in the “Are you Ready for More” activity, which uses both cylinder and cone volume formulas.
The cooldown in Activity 15.4 also asks students to calculate volumes with both formulas.
Lesson 16 – Finding Cone Dimensions
What would you anticipate students having difficulty with as they work from a given volume and height to find the radius of a cone? Whatever you thought of ( undoing, the fraction, the pi, the squared term), it is addressed in advance in the warm up- don’t skip it!
16.1 Number Talk: Thirds
Do this activity with books closed, revealing and discussing equations one at a time.
If number talks are still new to you, here is a Number Talk Cheat Sheet with handy prompts for both you and students as you adjust to the routine.
16.2, 16.3, and 16.4
These activities mirror the work in lesson 14. All tips listed there apply in this lesson as well.
Community created resources Lesson 16:
Shout out to Rachel H who created a great review worksheet for volume formulas, similar to the charts in 14.3 and 16.3. Since this includes spheres, you may choose to use this as part of your unit review.
Next up in the unit –
Lessons 17 and 18 connect volume patterns to functions and give students an opportunity to identify linear and non-linear functions. Time crunched teachers could consider combining these lessons.
Lesson 19 – a beautiful lesson that builds an intuition for the volume formula of a sphere. Don’t cut this.
Lesson 20 finishes and formalizes the volume of a sphere work. Could be shortened or combined with 21.3 and 21.4 to review the unit.
Lesson 22 – wraps the unit nicely, connecting functions and their graphs with the volume learning. Could be skipped if time is an issue.